Thermal convection in nanofluids for peristaltic flow in a nonuniform channel

A magneto couple stress nanofluid flow along with double diffusive convection is presented for peristaltic induce flow through symmetric nonuniform channel. A comprehensive mathematical model is scrutinized for couple stress nanofluid magneto nanofluids and corresponding equations of motions are tackled by applying small Reynolds and long wavelength approximation in viewing the scenario of the biological flow. Computational solution is exhibited with the help of graphical illustration for nanoparticle volume fraction, solutal concentration and temperature profiles in MATHEMTICA software. Stream function is also computed numerically by utilizing the analytical expression for nanoparticle volume fraction, solutal concentration and temperature profiles. Whereas pressure gradient profiles are investigated analytically. Impact of various crucial flow parameter on the pressure gradient, pressure rise per wavelength, nanoparticle volume fraction, solutal concentration, temperature and the velocity distribution are exhibited graphically. It has been deduced that temperature profile is significantly rise with Brownian motion, thermophoresis, Dufour effect, also it is revealed that velocity distribution really effected with strong magnetic field and with increasing non-uniformity of the micro channel. The information of current investigation will be instrumental in the development of smart magneto-peristaltic pumps in certain thermal and drug delivery phenomenon.


List of symbols
www.nature.com/scientificreports/ of each term. Tackling such bio-mathematical problems are important to modernize the diagnostic processes of several issues arise in peristaltic phenomenon. Further physiological impacts are explored for the biological flow of rheological fluid through inclined geometries by [29][30][31][32][33] . Hayat et al. 34 elaborated the effect of thermal radiation along with MHD for the Jeffrey fluid. A mathematical investigation for the peristaltic flow of couple-stress fluid in a transverse non-symmetric channel with bon-isothermal scenario has been reported 35 . One may find related heat transfer analysis for peristaltically induce flow for couple stress fluid 21,36,37 .

Mathematical interpretation of physical problem
Let us emphasis on the flow dynamics of electrically conducting couple stress fluid in a nonuniform channel in an incompressible MHD flow. Waves pass alongside the channel walls, causing flow. Assume we have a rectangular coordinate system with the X-axis aligned with wave propagation and the Y-axis parallel to it. The induced magnetic field is led by a continuous magnetic area of strength acting in a transverse direction. The magnetic field in general is defined as (Fig. 1) The following is the geometrical description of the physical problem 21-33,33-37 : with a(X) = a 0 + a 1 X. Within equation the parameters ,a 0 , t, a , b, c, represents wavelength, half width at inlet, time, half breadth of channel, wave amplitude, speed of the wave and respectively.
The continuity, momentum, energy, concentration, and nanoparticle volume fraction equations 33,34 are: www.nature.com/scientificreports/ In order to further simplified the flow analysis, the following transformations would be used to examine the flow from laboratory frame of situation to wave frame scenario.
Making use of underneath transformations one may have.
Making use of above quantities in Eqs. (2-7) along with long wavelength and low Reynolds numbers we get. (3) (4) where m 1 non dimensional width of the inlet, α is the non-uniform width 22,23 of the channel. The expression for current density and axial induced magnetic field is defined as.
To explore pressure rise per wavelength we may have.

Solution of the problem
Equations (11)(12)(13) are tackled numerically in Mathematica, whereas the Eq. (15) after capitalizing exact expressions from Eqs. (21)(22)(23) is also solved computationally. The expression for nanoparticle volume fraction is obtained from Eq. (13) as, Similarly, the expression for solutal concentration is obtained from Eq. (12) as, By utilizing Eq. (11) the temperature expression is obtained as, The constants c1 through c6 are determined using boundary conditions (16,17). The following are the values of these constants:  Figures 2 and 3 elucidate the influence of non-uniformity parameter m1 and thermophoresis parameter N t on pressure gradient dp/dx. It is revealed from Fig. 2 that an incremental increase in non-uniformity parameter m1, the pressure gradient diminishes      www.nature.com/scientificreports/ asymptotically. From physical point of view, it is natural phenomenon since the channel width is enhanced naturally pressure profile will decrease. From Fig. 3, we observed that as the value of thermophoresis parameter N t enhances, reduction in the pressure gradient is observed. Physically thermophoresis phenomenon results into higher molecular movement which results into a decay in pressure gradient. Figures 4 and 5 exhibit the impacts of non-uniformity parameter m1 and thermophoresis parameter N t on the pumping mechanism. It is observed that pumping phenomenon is greatly influenced by these parameters, pressurize profile is depreciated in region y ∈ [−2, 1] and is surges in y ∈ [−1, 4] as non-uniformity parameter m1 is enhanced. Further, it is reported that pressure rise is compactly surges as thermophoresis parameter N t is strengthened. Figures 6 and 7 show the effects     . It is due to the fact that magnetic Reynolds number is the ratio of induction and diffusion, it provides a guess of the relative impacts of induction due to magnetic field due to the dynamics of a conducting medium. Figure 7 shows opposite trends for positive values of magnetic field M. Since Hartmann number provides an estimate of the relative significance of drag forces which are generated from magnetic induction and viscous forces during the flow. It may be deduced that axial induced magnetic field distribution is declined initially and then turn around is seen. In order to observe the influences of magnetic Reynolds number R m and   . Current density is referred as charge per unit time that flows within some specified region. It is quite evident from the Fig. 8 that the magnetic Reynolds number reinforce current density distribution. Further, as narrated above the magnetic number retard the flow, in the similar manner a declined in the current density distribution is observed as magnetic field is become stronger (Fig. 9). Figures 10, 11, 12 and 13 have been prepared in order to investigate the phenomenon of Brownian motion N b , thermophoresis parameter N t , Soret parameter N CT and Dufour parameter N TC on the temperature profile θ. It is noticed that temperature profile enhances with an incremental change in Brownian motion N b , thermophoresis parameter N t , Soret parameter N CT and Dufour parameter N TC . The qualitative    Fig. 18, it is noted that as the value of N b increases, the magnitude of nanoparticle fraction increases in magnitude whereas opposite trend is noted for N t , N CT and N TC . It is also observed maximum variation in nanoparticle fraction is noted near the lower part of

Conclusion
A mathematical model has been presented for couple stress magneto nanofluids and corresponding equations of motions are handled by applying low Reynolds and long wavelength approximation in viewing the scenario of the physical flow. Computational solution has been explored for nanoparticle volume fraction, solutal concentration and temperature profiles in MATHEMTICA software. The crux of the current study may be interpreted as: • The pressure gradient decreases by enhancing the values of thermophoresis and non-uniformity parameter.